We prove some uniqueness theorems for series in general Franklin systems. In particular, for series in the classical Franklin system our result asserts that if the partial sums $S_{n_i}(x)=\sum_{k=0}^{n_i}a_kf_k(x)$ of a Franklin series $\sum_{k=0}^{\infty}a_kf_k(x)$ converge in measure to an integrable function $f$ and $\sup_i|S_{n_i}(x)|<\infty$, for $x\notin B$, where $B$ is some countable set and $\sup_i(n_i/n_{i-1})<\infty$, then this is the Fourier-Franklin series of $f$. Bibliography: 29 titles.